Generic Expansions by a Reduct

Generic expansion of an abelian variety by a subgroup

“…We denote by acl T (X ) the algebraic closure of X ⊂ A in the sense of T . Then using [10, Theorem 4.1, Corollary 4.3] as in the proof of [10,Theorem 5.28], we get the following corollary.…”

“…We do not have the answer in the context of abelian varieties, however, if we allow A to be an affine algebraic group we get a negative answer. In [10], ACFG is the model-companion of an algebraically closed field of fixed positive characteristic with a predicate for an additive subgroup. In the context of this note, it is T G with T = ACF p for p > 0 and A = G a .…”

“…In particular, if A = G n a and the ambient field is of characteristic 0, then the stabiliser of G is Z (or Q if G is assumed divisible) and the model-companion of T G does not exist (cf. [10,Subsection 5.6] for the case n = 1). Conversely, if the endomorphism ring is definable and infinite, then it is an algebraically closed field, hence the group A is a K-vector space, hence isomorphic to G n a .…”

“…An irreducible quasi-variety is a subset of A n of the form V ∩ O where V is an irreducible variety and The result of Proposition 2.5 is, in general, the hard step in order to axiomatise existentially closed models of T G ; it corresponds to hypothesis (H 4 ) in [10]. We gather now the data we have collected concerning the theory T and its reduct T A , when End(A) = Z:…”

“…From [10,Theorem 4.1] in order to prove that T G is NSOP 1 and to get the description of Kim-independence, it is sufficient to prove that if X, Y, Z are acl T -closed, containing a model A of T , and if Z | T A X, Y , then acl T (XZ), acl T (Y Z) ∩ acl T (XY ) = X, Y . Let X, Y, Z be as such, and let w ∈ acl T (XZ), acl T (Y Z) ∩ acl T (XY ).…”